Precalculus Examples

$x + y + 4 z = 1$ , $3 x + 2 y + 11 z = 1$ , $x + 3 z = - 1$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Subtract $y$ from both sides of the equation.

$x + 4 z = 1 - y$

$3 x + 2 y + 11 z = 1$

$x + 3 z = - 1$

Step 1.2

Subtract $4 z$ from both sides of the equation.

$x = 1 - y - 4 z$

$3 x + 2 y + 11 z = 1$

$x + 3 z = - 1$

$x = 1 - y - 4 z$

$3 x + 2 y + 11 z = 1$

$x + 3 z = - 1$

Step 2

Replace all occurrences of $x$ with $1 - y - 4 z$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $3 x + 2 y + 11 z = 1$ with $1 - y - 4 z$ .

$3 (1 - y - 4 z) + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $3 (1 - y - 4 z) + 2 y + 11 z$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$3 \cdot 1 + 3 (- y) + 3 (- 4 z) + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.2.1.1.2

Simplify.

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Step 2.2.1.1.2.1

Multiply $3$ by $1$ .

$3 + 3 (- y) + 3 (- 4 z) + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.2.1.1.2.2

Multiply $- 1$ by $3$ .

$3 - 3 y + 3 (- 4 z) + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.2.1.1.2.3

Multiply $- 4$ by $3$ .

$3 - 3 y - 12 z + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

$3 - 3 y - 12 z + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

$3 - 3 y - 12 z + 2 y + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.2.1.2

Simplify by adding terms.

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Step 2.2.1.2.1

Add $- 3 y$ and $2 y$ .

$3 - y - 12 z + 11 z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.2.1.2.2

Add $- 12 z$ and $11 z$ .

$3 - y - z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$x + 3 z = - 1$

Step 2.3

Replace all occurrences of $x$ in $x + 3 z = - 1$ with $1 - y - 4 z$ .

$(1 - y - 4 z) + 3 z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 2.4

Simplify the left side.

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Step 2.4.1

Add $- 4 z$ and $3 z$ .

$1 - y - z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$1 - y - z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$1 - y - z = - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3

Solve for $y$ in $1 - y - z = - 1$ .

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Step 3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.1.1

Subtract $1$ from both sides of the equation.

$- y - z = - 1 - 1$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.1.2

Add $z$ to both sides of the equation.

$- y = - 1 - 1 + z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.1.3

Subtract $1$ from $- 1$ .

$- y = - 2 + z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$- y = - 2 + z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.2

Divide each term in $- y = - 2 + z$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- y = - 2 + z$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 2}{- 1} + \frac{z}{- 1}$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 2}{- 1} + \frac{z}{- 1}$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 2}{- 1} + \frac{z}{- 1}$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$y = \frac{- 2}{- 1} + \frac{z}{- 1}$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Divide $- 2$ by $- 1$ .

$y = 2 + \frac{z}{- 1}$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.2.3.1.2

Move the negative one from the denominator of $\frac{z}{- 1}$ .

$y = 2 - 1 \cdot z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 3.2.3.1.3

Rewrite $- 1 \cdot z$ as $- z$ .

$y = 2 - z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$y = 2 - z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$y = 2 - z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$y = 2 - z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

$y = 2 - z$

$3 - y - z = 1$

$x = 1 - y - 4 z$

Step 4

Replace all occurrences of $y$ with $2 - z$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $3 - y - z = 1$ with $2 - z$ .

$3 - (2 - z) - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $3 - (2 - z) - z$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$3 - 1 \cdot 2 + z - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2.1.1.2

Multiply $- 1$ by $2$ .

$3 - 2 + z - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2.1.1.3

Multiply $- - z$ .

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Step 4.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$3 - 2 + 1 z - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2.1.1.3.2

Multiply $z$ by $1$ .

$3 - 2 + z - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

$3 - 2 + z - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

$3 - 2 + z - z = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Combine the opposite terms in $3 - 2 + z - z$ .

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Step 4.2.1.2.1.1

Subtract $z$ from $z$ .

$3 - 2 + 0 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2.1.2.1.2

Add $3 - 2$ and $0$ .

$3 - 2 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

$3 - 2 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.2.1.2.2

Subtract $2$ from $3$ .

$1 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

$1 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

$1 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

$1 = 1$

$y = 2 - z$

$x = 1 - y - 4 z$

Step 4.3

Replace all occurrences of $y$ in $x = 1 - y - 4 z$ with $2 - z$ .

$x = 1 - (2 - z) - 4 z$

$1 = 1$

$y = 2 - z$

Step 4.4

Simplify the right side.

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Step 4.4.1

Simplify $1 - (2 - z) - 4 z$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$x = 1 - 1 \cdot 2 + z - 4 z$

$1 = 1$

$y = 2 - z$

Step 4.4.1.1.2

Multiply $- 1$ by $2$ .

$x = 1 - 2 + z - 4 z$

$1 = 1$

$y = 2 - z$

Step 4.4.1.1.3

Multiply $- - z$ .

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Step 4.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 - 2 + 1 z - 4 z$

$1 = 1$

$y = 2 - z$

Step 4.4.1.1.3.2

Multiply $z$ by $1$ .

$x = 1 - 2 + z - 4 z$

$1 = 1$

$y = 2 - z$

$x = 1 - 2 + z - 4 z$

$1 = 1$

$y = 2 - z$

$x = 1 - 2 + z - 4 z$

$1 = 1$

$y = 2 - z$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $2$ from $1$ .

$x = - 1 + z - 4 z$

$1 = 1$

$y = 2 - z$

Step 4.4.1.2.2

Subtract $4 z$ from $z$ .

$x = - 1 - 3 z$

$1 = 1$

$y = 2 - z$

$x = - 1 - 3 z$

$1 = 1$

$y = 2 - z$

$x = - 1 - 3 z$

$1 = 1$

$y = 2 - z$

$x = - 1 - 3 z$

$1 = 1$

$y = 2 - z$

$x = - 1 - 3 z$

$1 = 1$

$y = 2 - z$

Step 5

Remove any equations from the system that are always true.

$x = - 1 - 3 z$

$y = 2 - z$